Abstract:
In this paper, we consider the equations of Kaup system kind with a self-consistent source in the class of periodic functions. We discuss the complete integrability of the considered nonlinear system of equations, which is based on the transformation to the spectral data of an associated quadratic pencil of Sturm–Liouville equations with periodic coefficients. In particular, Dubrovin-type equations are derived for the time-evolution of the spectral data corresponding to the solutions of equations of Kaup system kind with self-consistent source in the class of periodic functions. Moreover, it is shown that spectrum of the quadratic pencil of Sturm–Liouville equations with periodic coefficients associated with considering nonlinear system does not depend on time. In a one-gap case, we write the explicit formulae for solutions of the problem under consideration expressed in terms of the Jacobi elliptic functions. We show that if p0(x)p0(x) and q0(x)q0(x) are real analytical functions, the lengths of the gaps corresponding to these coefficients decrease exponentially. The gaps corresponding to the coefficients p(x,t)p(x,t) and q(x,t)q(x,t) are same. This implies that the solutions of considered problem p(x,t)p(x,t) and q(x,t)q(x,t) are real analytical functions in xx.
Keywords:
equations of Kaup system kind, quadratic pencil of Sturm–Liouville equations, inverse spectral problem, trace formulas, periodical potential.
The authors express their gratitude to Prof. Aknazar Khasanov (Samarkand State University, Uzbekistan) for discussion and valuable advice, as well as to the International Erasmus+Program KA106-2, Keele University, UK.
Citation:
A. B. Yakhshimuratov, B. A. Babajanov, “Integration of equations of Kaup system kind with self-consistent source in class of periodic functions”, Ufa Math. J., 12:1 (2020), 103–113
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\paper Integration of equations of Kaup system kind with self-consistent source in class of periodic functions
\jour Ufa Math. J.
\yr 2020
\vol 12
\issue 1
\pages 103--113
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\crossref{https://doi.org/10.13108/2020-12-1-103}
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Linking options:
https://www.mathnet.ru/eng/ufa506
https://doi.org/10.13108/2020-12-1-103
https://www.mathnet.ru/eng/ufa/v12/i1/p104
This publication is cited in the following 10 articles:
B.A. Babajanov, Sh.O. Sadullaev, M.M. Ruzmetov, “Integration of the Kaup–Boussinesq system via inverse scattering method”, Partial Differential Equations in Applied Mathematics, 11 (2024), 100813
B. A. Babajanov, F. B. Abdikarimov, F. U. Sulaymonov, “On the Integration of the Hierarchy of the Kaup–Boussinesq System with a Self-Consistent Source”, Lobachevskii J Math, 45:7 (2024), 3233
B. A. Babajanov, A. Sh. Azamatov, R. B. Atajanova, “Integration of the Kaup–Boussinesq system with time-dependent coefficients”, Theoret. and Math. Phys., 216:1 (2023), 961–972
Bazar Babajanov, Michal Fečkan, Aygul Babadjanova, “On the Differential-Difference Sine-Gordon Equation with an Integral Type Source”, Mathematica Slovaca, 73:6 (2023), 1499
B. A. Babajanov, A. K. Babadjanova, A. Sh. Azamatov, “Integration of the differential–difference sine-Gordon equation with a self-consistent source”, Theoret. and Math. Phys., 210:3 (2022), 327–336
Aknazar B. Khasanov, Bazar A. Babajanov, Dilshod O. Atajonov, “On the integration of the periodic Camassa–Holm equation with a self-consistent source”, Zhurn. SFU. Ser. Matem. i fiz., 15:6 (2022), 785–796
B. A. Babajanov, D. O. Atajonov, “On the integration of the periodical Camassa–Holm equation with an integral type source”, Russian Math. (Iz. VUZ), 66:11 (2022), 1–11
Bazar Babajanov, Fakhriddin Abdikarimov, “Exact Solutions of the Nonlinear Loaded Benjamin-Ono Equation”, WSEAS TRANSACTIONS ON MATHEMATICS, 21 (2022), 666
B. A. Babajanov, M. M. Ruzmetov, “On the construction and integration of a hierarchy for the periodic Toda lattice with a self-consistent source”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 38 (2021), 3–18
A. Yakhshimuratov, T. Kriecherbauer, B. Babajanov, “On the Construction and Integration of a Hierarchy for the Kaup System with a Self-Consistent Source in the Class of Periodic Functions”, Z. mat. fiz. anal. geom., 17:2 (2021), 233